feat: Implement polynomial exponential functions with NTT support

Author: weilycoder

test/exp_of_formal_power_series.test.cpp

@@ -0,0 +1,24 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/exp_of_formal_power_series"
+
+#include "../weilycoder/poly/elementary_func/exponential.hpp"
+#include <cstdint>
+#include <iostream>
+#include <vector>
+using namespace std;
+using namespace weilycoder;
+
+int main() {
+  cin.tie(nullptr)->sync_with_stdio(false);
+  cin.exceptions(cin.failbit | cin.badbit);
+  size_t n;
+  cin >> n;
+
+  vector<uint64_t> a(n);
+  for (size_t i = 0; i < n; ++i)
+    cin >> a[i];
+
+  auto res = ntt_exponential<998244353>(a, n);
+  for (size_t i = 0; i < n; ++i)
+    cout << res[i] << " \n"[i + 1 == n];
+  return 0;
+}

weilycoder/poly/elementary_func/exponential.hpp

@@ -0,0 +1,69 @@
+#ifndef WEILYCODER_POLY_ELEMENTARY_FUNC_EXPONENTIAL_HPP
+#define WEILYCODER_POLY_ELEMENTARY_FUNC_EXPONENTIAL_HPP
+
+/**
+ * @file exponential.hpp
+ * @brief Polynomial Exponential Functions
+ */
+
+#include "../../number_theory/mod_utility.hpp"
+#include "logarithm.hpp"
+#include <vector>
+
+namespace weilycoder {
+/**
+ * @brief Compute the exponential of a polynomial modulo x^n.
+ * @tparam T Coefficient type.
+ * @tparam PolyMultiplyFunc Type of the multiplication function.
+ * @tparam AddFunc Type of the addition function for coefficients.
+ * @tparam SubtractFunc Type of the subtraction function for coefficients.
+ * @tparam MultiplyFunc Type of the multiplication function for coefficients.
+ * @tparam InverseFunc Type of the inversion function for coefficients.
+ * @param poly Coefficients of the polynomial.
+ * @param n Degree up to which the exponential is computed (result size will be n).
+ * @param multiply Function to multiply two polynomials.
+ * @param number_add Function to add two coefficients.
+ * @param number_sub Function to subtract two coefficients.
+ * @param number_mul Function to multiply two coefficients.
+ * @param number_inv Function to compute the multiplicative inverse of a coefficient.
+ * @return Coefficients of the exponential polynomial.
+ */
+template <typename T, typename PolyMultiplyFunc, typename AddFunc,
+          typename SubtractFunc, typename MultiplyFunc, typename InverseFunc>
+std::vector<T> exponential(const std::vector<T> &poly, size_t n,
+                           PolyMultiplyFunc multiply, AddFunc number_add,
+                           SubtractFunc number_sub, MultiplyFunc number_mul,
+                           InverseFunc number_inv) {
+  if (!poly.empty() && poly[0] != T(0))
+    throw std::invalid_argument(
+        "Exponential is only defined for polynomials with constant term 0.");
+  std::vector<T> result = {T(1)};
+  for (size_t len = 1; len < n; len <<= 1) {
+    size_t m = std::min(n, len << 1);
+    auto temp = logarithm(result, m, multiply, number_sub, number_mul, number_inv);
+    temp.front() = number_sub(T(1), temp.front());
+    for (size_t i = 1; i < temp.size(); ++i)
+      temp[i] = number_sub(T(0), temp[i]);
+    for (size_t i = 0; i < std::min(poly.size(), m); ++i)
+      temp[i] = number_add(temp[i], poly[i]);
+    result = poly_mul(result, temp, m, multiply);
+  }
+  return result;
+}
+
+/**
+ * @brief Compute the exponential of a polynomial modulo x^n using NTT
+ *        under a given modulus.
+ * @tparam mod Modulus for NTT.
+ * @param poly Coefficients of the polynomial.
+ * @param n Degree up to which the exponential is computed (result size will be n).
+ * @return Coefficients of the exponential polynomial.
+ */
+template <uint64_t mod>
+std::vector<uint64_t> ntt_exponential(const std::vector<uint64_t> &poly, size_t n) {
+  return exponential<uint64_t>(poly, n, ntt_convolve<mod>, mod_add<mod>, mod_sub<mod>,
+                               mod_mul<mod>, mod_inv<mod>);
+}
+} // namespace weilycoder
+
+#endif